Problem: Simplify and expand the following expression: $ \dfrac{5}{2p - 6}+ \dfrac{3}{3p - 6}+ \dfrac{5}{p^2 - 5p + 6} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{5}{2p - 6} = \dfrac{5}{2(p - 3)}$ We can factor a $3$ out of denominator in the second term: $ \dfrac{3}{3p - 6} = \dfrac{3}{3(p - 2)}$ We can factor the quadratic in the third term: $ \dfrac{5}{p^2 - 5p + 6} = \dfrac{5}{(p - 3)(p - 2)}$ Now we have: $ \dfrac{5}{2(p - 3)}+ \dfrac{3}{3(p - 2)}+ \dfrac{5}{(p - 3)(p - 2)} $ The least common multiple of the denominators is: $ 6(p - 3)(p - 2)$ In order to get the first term over $6(p - 3)(p - 2)$ , multiply by $\dfrac{3(p - 2)}{3(p - 2)}$ $ \dfrac{5}{2(p - 3)} \times \dfrac{3(p - 2)}{3(p - 2)} = \dfrac{15(p - 2)}{6(p - 3)(p - 2)} $ In order to get the second term over $6(p - 3)(p - 2)$ , multiply by $\dfrac{2(p - 3)}{2(p - 3)}$ $ \dfrac{3}{3(p - 2)} \times \dfrac{2(p - 3)}{2(p - 3)} = \dfrac{6(p - 3)}{6(p - 3)(p - 2)} $ In order to get the third term over $6(p - 3)(p - 2)$ , multiply by $\dfrac{6}{6}$ $ \dfrac{5}{(p - 3)(p - 2)} \times \dfrac{6}{6} = \dfrac{30}{6(p - 3)(p - 2)} $ Now we have: $ \dfrac{15(p - 2)}{6(p - 3)(p - 2)} + \dfrac{6(p - 3)}{6(p - 3)(p - 2)} + \dfrac{30}{6(p - 3)(p - 2)} $ $ = \dfrac{ 15(p - 2) + 6(p - 3) + 30} {6(p - 3)(p - 2)} $ Expand: $ = \dfrac{15p - 30 + 6p - 18 + 30}{6p^2 - 30p + 36} $ $ = \dfrac{21p - 18}{6p^2 - 30p + 36}$ Simplify: $ = \dfrac{7p - 6}{2p^2 - 10p + 12}$